There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as (locally) shortest curves (Euler-Lagrange, Pontyagin and Hamilton) and three definitions of geodesics as straightest curves (d'Alembert , Levi-Civita-Schouten and Cartan-Tanaka ), used in nonholonomic mechanics and discuss their interrelations. We consider a big class of sub-Riemannian manifolds associated with principal bundle over a Riemannian manifolds, for which shortest geodesics coincides with straightest geodesics. Using the geometry of flag manifolds, we describe some classes of compact homogeneous sub-Riemannian manifolds (including contact sub-Riemannian manifolds and symmetric sub-Riemannian manifolds) where straightest geodesics coincides with shortest geodesics. Construction of geodesics in these cases reduces to description of Riemannian geodesics of the Riemannian homogeneous manifold or left-invariant metric on a Lie group.

中文翻译：

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亚黎曼几何中最短和最直的测地线

黎曼测地线有许多等效的定义。它们自然地推广到亚黎曼流形，但变得不等价。我们回顾了亚黎曼流形的测地线的不同定义以及它们之间的相互关系。我们回顾了测地线的三个变分定义为（局部）最短曲线（欧拉-拉格朗日、庞蒂亚金和汉密尔顿）和测地线的三个最直曲线定义（d'Alembert、Levi-Civita-Schouten 和 Cartan-Tanaka），用于非完整力学并讨论它们的相互关系。我们考虑了一大类与黎曼流形上的主丛相关的亚黎曼流形，其中最短测地线与最直测地线重合。使用标志流形的几何形状，我们描述了一些紧凑齐次亚黎曼流形（包括接触亚黎曼流形和对称亚黎曼流形），其中最直的测地线与最短的测地线重合。在这些情况下，测地线的构造简化为黎曼齐次流形或李群上的左不变度量的黎曼测地线的描述。